Probably the most famous non-Christian source used as "evidence" for a historic Jesus, is the Roman senator, consul, speaker, and historian Cornelius Tacitus ( 20 - 117 AD). In a passage in his "Annales, book 15, verse 44" from the year 115 AD concerning the Christians, he mentions the name "Christ" as the subject for the Christian's cult and worship:
"Christus, the founder of the name, had undergone the death penalty in the reign of Tiberius, by sentence of Pontius Pilate, and the pernicious [or wicked] superstition [Christianity] was checked for a moment, only to break out once more, not merely in Judea, the home of the disease, but in the capital [Rome] itself, where all things horrible or shameful in the world collect and find a vogue."
This is mere hearsay; it's by no means an eyewitness report or useable as evidence of a historical Jesus-figure.
Then we have Tranquillus Gaius Suetonius ( 69 - 140 AD), a Roman historian and the personal secretary of emperor Hadrian.
Suetonius also mentions the name Chrestus as the subject of the Christians worship. "Since the Jews constantly made disturbances at the instigation of Chrestus (Claudius) expelled them from Rome". ("Judaeos, impulsore Chresto, assidue tumultuantes (Claudius) Roma expulit".) This is also mere hearsay, and not any kind of suitable evidence for the existence of a historical Jesus.
So is also the information by Pliny the Younger, Roman governor in Asia Minor around AD 110. In a letter to emperor Trajan, he asks what to do with the Christians who "sing responsively a hymn to Christ as to a god". Not exactly an eyewitness report of Jesus, and surely not hard evidence of anything else that Christians also sang hymns around 110 AD.
Josephus Flavius (born AD 37 or 38), a Jewish historian, does not mention any Jesus in his Jewish Antiquities, the history of the Jews from the beginning of time until the time of emperor Nero (published ca AD 93). Josephus mentions, among others, Pontius Pilate, John the Baptist and king Herod, and numerous events of minor and major political, religious and economical interests in the area. But he does not seem to know of any Jesus.
Then, in the third century, an earlier unknown addition to the Jewish Antiquities miraculously emerged, the so called "Testimonium Flavianum". Here Josephus suddenly witness Christ, and becomes a Christian. The problem is that this text is a forgery! Even parts of the Catholic Church acknowledges this. The forgery was probably written by the infamous bishop (and famous Church historian) Eusebius of Caesarea (ca AD 265-330). He forged a lot of texts in his time.
What about Christ's contemporaries?
None of the literate contemporaries of Jesus know anything of him. The Jewish historian Justus of Tiberia who lived at the time of Jesus, do not know of him. (Tiberia was a place not far from Capernaum which Jesus often visited, according to the Bible).
The Jewish scholar and leader of the Jewish society in Alexandria, Philon of Alexandria (around AD 30 - 45) does not mention any Jesus anywhere in his texts. Philon was a famous scholar of the Old Testament and had deep knowledge of the Jewish cults of his time. He died ca. AD 50.
There is thus no real historical evidence of a historical Jesus. One would suppose that, a character like Jesus who according to the gospels raised the dead, healed the sick and annoyed both the Jewish establishment and the mighty Romans to such a degree that they finally had to execute him, one should think such a character would make it into at least some contemporary historical texts. Nope. No record.
In view of the evidence the only honest conclusion is that the Gospel's Jesus never existed.
That there once lived a wannabe-Messiah named Joshua (greek: Jesus) in the first century Palestine is more than probable. Roman sources tell of dozens of more or less religiously confused wannabe-Messiahs at the time, and Joshua was a very common Jewish name. But this could not be the Gospel's Jesus, not the Son of God, raising the dead, healing the sick, annoying the establishment, executed as a criminal, and then finally flapping away to heaven. All that stuff is pure mythical, and blatantly stolen from older pagan cults by the anonymous Gospel-authors.